Editing Magnetostatics (section)

==Applications==

===Magnetostatics as a special case of Maxwell's equations===
Starting from [[Maxwell's equations]] and assuming that charges are either fixed or move as a steady current <math>\mathbf{J}</math>, the equations separate into two equations for the [[electric field]] (see [[electrostatics]]) and two for the [[magnetic field]].<ref>[https://feynmanlectures.caltech.edu/II_13.html The Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics]</ref> The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.
{| class="wikitable"
|- style="background-color: #aaddcc;"
! rowspan=2 | Name
! colspan=2 | Form
|-
! [[Partial differential equation|Differential]]
! [[Integral]]
|-
| [[Gauss's law for magnetism|Gauss's law <br/>for magnetism]]
| <math>\mathbf{\nabla} \cdot \mathbf{B} = 0</math> 
| <math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{S} = 0</math>
|-
| [[Ampère's law]]
| <math>\mathbf{\nabla} \times \mathbf{H} = \mathbf{J}</math>
| <math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{\mathrm{enc}}</math>
|}

Where ∇ with the dot denotes [[divergence]], and '''B''' is the [[magnetic flux density]], the first integral is over a surface <math> S</math> with oriented surface element <math> d\mathbf{S}</math>. Where ∇ with the cross denotes [[Curl (mathematics)|curl]], '''J''' is the [[current density]] and {{math|'''H'''}} is the [[magnetic field intensity]], the second integral is a line integral around a closed loop <math> C</math> with line element <math>\mathbf{l}</math>. The current going through the loop is <math> I_\text{enc}</math>.

The quality of this approximation may be guessed by comparing the above equations with the full version of [[Maxwell's equations]] and considering the importance of the terms that have been removed. Of particular significance is the comparison of the <math> \mathbf{J}</math> term against the <math> \partial \mathbf{D} / \partial t</math> term. If the <math>\mathbf{J}</math> term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.

===Re-introducing Faraday's law===
A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term <math> \partial \mathbf{B} / \partial t</math>. Plugging this result into [[Faraday's law of induction|Faraday's Law]] finds a value for <math> \mathbf{E}</math> (which had previously been ignored). This method is not a true solution of [[Maxwell's equations]] but can provide a good approximation for slowly changing fields.{{Citation needed|date=October 2010}}